![]() ![]() Let us use this formula to represent the resolvent matrix as a series. We assume that the series on the right-hand side of this equation converges. Next, we will show that the state transition matrix is actually equal to the matrix exponential of. That is the state of the system at the time instant is calculated by multiplying the state transition matrix by the initial state. On the other hand, the matrixĬonsequently, the equation ( 8) can be written as follows The matrix is called the resolvent matrix of the matrix. Where is the notation for the inverse Laplace transform. Let us apply the inverse Laplace transform to the last equation. Let us apply the Laplace transform to the system ( 16). Let us assume that the initial condition of the system is given by. Where is the state vector and is the system matrix. Let us consider a linear dynamical system Matrix exponential, dynamical systems, and control theory We answer these questions in the next section.
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